 Hi, I'm Zor. Welcome to Unisor Education. The previous lecture was about inertial reference frames. This lecture is about non-inertial frame references, obviously. Now, this lecture and all other lectures I'm doing right now are part of the Physics 14 course of physics which is part of the Unisor.com educational website. I suggest you to watch this lecture from the website rather than from, let's say, YouTube, because the website contains very detailed notes and it will contain some problems, exams, etc. The course of Physics 14 is just in the very beginning. Before that, there is a completed course of Mass 14s and then there is also the Civics course, US Law 14s, and it's all on this website. It's all free-for-all, no advertisement, so please use it. Now, let's talk about non-inertial reference frame. Alright, now the first part of this lecture is about definition of what is a non-inertial frame of reference. Well, that's a very easy part. The frame of reference is the frame of reference which is not inertial. Now, we have to probably repeat what is inertial frame of reference. Well, we could say that in the, let's say, original formulation of this, it's about the system of coordinates where if the object does not experience any acting on it or the forces are balanced, either or. Actually, there is no unbalanced forces. Then it's supposed to stay, if it stays and moves with a constant velocity, if it moves. So, basically the first when it stays is part of the constant velocity with a velocity vector equals to zero vector. So, in any case we have this constancy of the velocity vector as the necessary condition for the frame of reference to be inertial if there are no unbalanced forces acting on the object. Well, basically all these words, all this formulation is brought into one specific law. It's called law of inertia, which we take as an axiom, basically, for any inertial system. So, basically inertial system is where the law of inertia is true and the law of inertia is what I was just talking about. If there are no unbalanced forces, the body will continue its movement with a constant velocity, maybe equal to zero. All relative to this inertial reference frame. And now, back to non-inertial, that's where the law of inertia is not held true. So, if we have some particular object in a frame of reference which is not moving with constant velocity vector, while there are no unbalanced forces acting upon it, then this system is non-inertial, basically. That's where it is. So, the law of inertia is not held true in this system, at least sometimes. Now, this is my first part of this lecture, which is very easy. As I was saying, non-inertial is not inertial. Now, to feel basically, maybe they don't exist, these non-inertial systems. Maybe every system of coordinates, whichever we can come up with, is inertial. Now, we started, if you remember, from the system which is related to seemingly immovable stars. And, well, basically to a certain degree of approximation, we can consider the system to be inertial, because the body, if it's somewhere in space, it's moving with a constant velocity where there are no gravitational fields or no any other forces acting upon it. And then, we have proven the theorem that if there is a system of coordinates which is uniformly moving relative to some inertial system, then it's also inertial. And that gives us a lot of inertial system. Now, my question is, do they exist these non-inertial systems? Here are a couple of examples. Okay. Let's consider my first example is this is my inertial system, where the law of inertia is true. So, if there is some kind of a body here, let's forget about z-coordinate. Let's consider that this body is on the x-y plane. So, there is this body and actually it can move on the x-axis. And let's say this is coordinate 1. So, it's 1, 0, 0, the coordinates of this particular point. Now, if there are no acting forces, unbalanced forces, and this is an inertial system, I have chosen this particular object, I put it on this particular place, no initial velocity into any direction. So, it stays in this place with coordinates 1, 0, 0. Well, let's put it x, y, z. So, the coordinates of this point is 1, 0, 0 independent law of time. It's always. Okay. Now, let's consider a different frame of reference. Here it is. Imagine frame of reference, which is, well, initially let's say this is u, it's not perpendicular enough. Well, that suggests it's fine. This is v, and this is w. So, they have the same origin, these two coordinate system. Now, my w-axis coincides with z-axis. Now, u and v are determining the plane, which is the same as x, y plane, right? But now, this system of coordinates, u, v, w, is not just stanging still, it's rotating around the axis w with certain angular speed w. So, how can we imagine it in the practical life? Well, for instance, you have a carousel. Now, if you are standing outside of the carousel, that's where you are. Now, I am standing on the carousel, and the carousel is rotating with certain speed. So, from my position, I'm standing still on the carousel, and I consider that the whole world is rotating around me, right? I mean, everything is relative. So, relative to me, this point you, in this case, as an observer, relative to me, which is standing on the carousel itself, you are rotating. So, what is, if my angular speed is omega, and let's say it's counterclockwise. Now, what happens with this particular point? Well, this point, from the perspective of UW coordinates, is rotating clockwise. Now, the coordinate system goes counterclockwise, so the point relative to this system will be clockwise. And how it will be rotating, basically? Well, if this is the angular speed, it means that at time t, it will turn by the angle, and if this is the radius equals to 1, so my coordinates, my U coordinate will be 1 which is the radius times cosine omega t, and we will be also 1, the radius times sine of omega t. So, this is my movement. Well, first of all, the point is not standing still, as in the XYZ coordinate where it's just standing still. Time doesn't really change the coordinate. In this case, time changes coordinate, because it goes in a circle. Not only that, it's a circular movement. Circular movement is not a straight line movement, so it's not like a movement with a constant velocity, which means our law of inertia is not holding true in this particular case. We have an object which experiences no acting upon its forces, and yet in this UVW coordinate system, it's going in a circle. That's the contradiction to the law of inertia, which means law of inertia is not held true to this in this particular frame of reference, which means this frame of reference is not inertial or non-inertial. Being not inertial means it's non-inertial. Okay, that's my first example. Just for you to feel that there are non-inertial systems. Now, let's consider a little bit more practical case. We all live on the planet Earth, and we are assuming that, well, we can conduct certain experiment, and we are thinking that, well, if I will just, you know, push let's say a scooter or something once in a straight line, it will go in a straight line, as if it is because of the law of inertia. Well, that's true, but it's not exactly true because the Earth is really rotating around its axis similarly to this, which means that the system of coordinates which is connected with the surface of the Earth is not exactly inertial system, and maybe the law of inertia is not working. Well, the answer is no, it's not working. That's absolutely true. However, considering that the Earth is so big and basically the radius is very big, and we are talking about a very small distance during which we are observing this particular thing. Small in a distance and small in time as well. So these parameters actually are contributing to our impression that if we have a system of coordinates related to one particular point on the planet Earth, then it's basically inertial. It's not, but the deviation from being inertial is really very, very small because the Earth is so big and the radius of rotation is very big. Similarly, if you have this particular radius and you take a particular piece of this, you really see the curvature. But if the radius is something like this and you have a similar piece of circumference, it's much more resembling the straight line than this one because the curvature in this case is greater and this case is less. So the curvature of Earth is very small on the surface and the rotation is one rotation in 24 hours. That's also quite a long period of time relative to our experiment which takes probably like 20 seconds. Well, not only that, the Earth is actually going around the Sun with a much bigger speed. It's like I don't remember exactly the number, something like 27 km per second or whatever it is. I mean, it's just huge. So again, considering this speed, but considering the radius it rotates around the Sun, all these characteristics of the movement of the Earth contribute to our impression that it's almost inertial. But again, in theory there are no inertial systems because everything is rotating around something in God knows what fashion. But again, to a certain approximation we can actually assume that certain systems like the system which is related to stars are inertial. Now let me talk about another example. So the rotation definitely gives you a non-inertial system. What else? Again, just as an example I'm not going to talk about all the possible non-inertial system. Just two examples. Okay, so we start with the same inertial system and a point P with co-ordination 1, 0, 0. Now my system which I will prove to be non-inertial, initially at moment T equals to 0, my u, v, w system is exactly the same as x, y, z. Which means this is u, this is v, and this is w. X's are exactly the same and origin is exactly the same. But then at moment T equals to 0, my u, v, w system starts moving. Moving to the right along the x-axis. So the next position is something like this. This would be v and this would be w. So it moves this way. Now what happens with my coordinates? Well, if this piece on which I moved is equal to a, my coordinate in the new system will be only this piece which is 1 minus a. Right? And then 0, 0. But if my movement is accelerating so the distance in the time t, the distance will be let's say a t square or something like this. That's if you remember how the distance looks in case you are accelerating. To be exact, if a is acceleration, the distance at moment t is this one. Right? If we start with 0 location and 0 initial speed that was in the previous lectures. So what does it mean? It means that my position in the u, v, w system would be u, v, w of t. It will be 1 minus a t square over 2. Which means it also will be accelerating. So as my system of coordinates goes to the right with acceleration a coordinate of the point p would be moving to the left with acceleration a. Well, it is acceleration a because the first derivative of this is minus 2 a t divided by 2 which is minus a t. And my second derivative which is, so this is v of t and the a of t is equal to minus a. Minus because it goes to the left relative to the coordinate system. If coordinate system goes to the right, the coordinate of this point goes to the left. That is why it is minus. So the second derivative is not equal to 0. Which means my point from the position of the u, v, w system goes to the left to the negative direction accelerating. Which is again against the law of inertia which says that if there are no forces, the velocity must be constant and that is why no acceleration. So this is another. So any system which accelerates in some way relative to inertial system is non-inertial. And in the previous example any system which rotates in some way relative to inertial system is not inertial itself. So these are two examples. Now let me just prove, well it's kind of similar to whatever I was just talking about but in more kind of formula type way. Let me prove that if you have two systems, x, y, z is inertial system and the position of the point is either constant or it's a movement with constant velocity vector. Now if my another system, my u, v, w system such that it moves relative to this system in such a way that the axis are parallel correspondingly parallel always. So it's like this would be my one system and this would be my another system and the movement is this. So if x, y, z system is inertial and u, v, w system is moving parallel to x, y, z with axis parallel to x, y, z but the origin is moving somewhere. So my point is that if this movement is not with a constant velocity then this system is not inertial. So this is the vector q in x, y, z system characterizes the location of the origin. So all we need is this function q which specifies location of the u, v, w system this point x, y, z coordinates. If this is not a constant velocity vector I mean if the first derivative is not constant basically. Now this is vector by the way. I didn't specify it. These are vectors. Position is a vector. So if my position of the origin of coordinate of the u, v, w in x, y, z coordinates. If it's not a constant velocity vector then this system is non-inertial. How can I prove it? Very simple. Let's take any point. So its location is in x, y, z system this one. And I know that the first derivative of this is in the absence of forces and considering this is inertial system the first derivative is a constant vector. Now let's talk about different implementation of this. If I will take this vector. Now this vector is coordinate of this point in u, v, w system so this is p, u, v, w of t. And this would be p, x, y, z of t. So we have a vector from origin of x, y, z to this point. The vector from origin of x, y, z to origin of u, v, w and from the origin of u, v, w to the point. Now obviously this vector is equal to this plus this. So from here to here this is this vector. It's the same as from here to here which is this plus from here to here which is this. Now this is a vector of position which has a constant velocity. So if we will take the first derivative we will have v, x, y, z of t which is constant equals to v u, v, w plus q prime. That's the speed of movement of the origin of the u, v, w system in x, y, z coordinates. Now if this is the constant because my body is not experiencing any acting force and these are velocity vector in the x, y, z inertial system of coordinates. If this is not a constant then this is not a constant. And since this is not constant it means that u, v, w is not inertial system. So it's non-inertial basically. So this is the way how we can prove certain things which means that we have an infinite number of inertial system and infinite number of non-inertial system. We just have to be very careful. Well considering my infinite number of inertial system is basically infinite number of systems which are almost inertial, we were talking about basically not having inertial system at all in a pure 100% mathematical sense. But in a certain approximation we know that there are many systems which can be considered inertial in our experiments, in our theoretical manipulation, etc. But many systems are obviously non-inertial. Significantly non-inertial. So there are inertial system which are actually non-inertial, non-inertial, but in an insignificant sense they are almost inertial. And then there are really non-inertial systems which manifest their non-inertiality very obviously. Okay, so inertial and non-inertial systems are very important. And most likely we will try to deal with anything, whatever we are dealing with, using inertial systems. Now whenever we are dropping the stone from the top of the tower of Pisa we are thinking that tower of Pisa represents inertial system and then we can basically calculate how the stone will fall etc. Whenever we are launching a projectile from some place on the surface of the earth, we are considering that the surface of the earth actually is the basis for some inertial system. And relative to this system we are calculating how our projectile is moving etc. So you just have to understand that there are non-inertial systems and most likely we will try to avoid them. That's basically my kind of final note in this particular lecture. Why don't you read exactly the notes for this lecture on Unizor.com. I was trying to write the notes as a textbook basically. So now you have the advantage of hearing whatever it is and also reading and sometimes they might actually contribute to each other these two sources of information. Alright, that's it for today. Thank you very much and good luck.